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A361321 Lexicographically earliest infinite sequence of distinct elements of A000469 such that, for n > 2, a(n) has a common factor with a(n-1) but not with a(n-2).

%I A361321 #11 Mar 11 2023 00:14:24
%S A361321 1,6,10,35,21,33,22,14,91,39,15,55,77,42,26,65,85,34,38,57,51,119,70,
%T A361321 30,69,161,133,95,110,46,299,143,66,58,145,105,78,62,155,115,138,74,
%U A361321 185,165,87,203,154,82,123,93,217,182,86,129,111,259,238,94,141,159
%N A361321 Lexicographically earliest infinite sequence of distinct elements of A000469 such that, for n > 2, a(n) has a common factor with a(n-1) but not with a(n-2).
%C A361321 This sequence is a variant of A360519 where we only consider nonprime squarefree numbers (A000469).
%C A361321 Theorem: a(1) = 1, a(2) = 6; thereafter, a(n) is the smallest nonprime squarefree number m not yet in the sequence such that
%C A361321 (i) gcd(m, a(n-1)) > 1,
%C A361321 (ii) gcd(m, a(n-2)) = 1, and
%C A361321 (iii) m does not divide a(n-1).
%C A361321 Conjecture: The sequence is a permutation of A000469.
%H A361321 Scott R. Shannon, <a href="/A361321/b361321.txt">Table of n, a(n) for n = 1..100000</a>
%H A361321 Scott R. Shannon, <a href="/A361321/a361321.png">White on black graph of first 50000 terms</a> [The green line is x = y]
%H A361321 Scott R. Shannon, <a href="/A361321/a361321_1.png">Image of the first 500000 terms in color</a>. The terms with a lowest prime factor of 2,3,5,7,9,11,13,17,19,>=23 are colored white, red, orange, yellow, green, blue, indigo, violet, gray respectively.
%H A361321 Rémy Sigrist, <a href="/A361321/a361321.gp.txt">PARI program</a>
%o A361321 (PARI) See Links section.
%Y A361321 Cf. A000469, A336957, A360519.
%K A361321 nonn
%O A361321 1,2
%A A361321 _Scott R. Shannon_, _Rémy Sigrist_ and _N. J. A. Sloane_, Mar 09 2023